Conic sections and their equations Master the definition, standard equation, and properties of the ellipse, and know the definitions, equations, and properties of the parabola and hyperbola.
Practice questions 1 过双曲线C C C x 2 a 2 − y 2 b 2 = 1 ( a > 0 , b > 0 ) \dfrac {x^{2}}{a^{2}}- \dfrac {y^{2}}{b^{2}} =1(a > 0 , b > 0) a 2 x 2 − b 2 y 2 = 1 ( a > 0 , b > 0 ) F F F 1 2 \dfrac {1}{2} 2 1 l l l 2 如图，两个椭圆x 2 25 + y 2 9 = 1 \dfrac{{x}^{2}}{25}+ \dfrac{{y}^{2}}{9}=1 25 x 2 + 9 y 2 = 1 y 2 25 + x 2 9 = 1 \dfrac{{y}^{2}}{25}+ \dfrac{{x}^{2}}{9}=1 25 y 2 + 9 x 2 = 1 C C C 3 若点A A A ( 4 , 2 ) (4,2) ( 4 , 2 ) F F F y 2 = 2 x y^{2}=2x y 2 = 2 x P P P ∣ P A ∣ + ∣ P F ∣ |PA|+|PF| ∣ P A ∣ + ∣ PF ∣ P P P ( ) () ( ) ( 1 , 2 ) (1,2) ( 1 , 2 ) ( 2 , 1 ) (2,1) ( 2 , 1 ) 4 已知三个实数2 2 2 m m m 8 8 8 x 2 m + y 2 2 = 1 \dfrac {x^{2}}{m}+ \dfrac {y^{2}}{2}=1 m x 2 + 2 y 2 = 1 ( ) () ( ) 2 2 \dfrac { \sqrt {2}}{2} 2 2 5 已知椭圆x 2 a 2 + y 2 b 2 = 1 ( a > b > 0 ) \dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0) a 2 x 2 + b 2 y 2 = 1 ( a > b > 0 ) F 1 F_{1} F 1 F 2 F_{2} F 2 F 1 F_{1} F 1 A ( x 1 , y 1 ) A(x_{1},y_{1}) A ( x 1 , y 1 ) 6 已知抛物线C C C x 2 = − 2 p y x^{2}=-2py x 2 = − 2 p y ( 2 , − 1 ) . ( 1 ) (2,-1).(1) ( 2 , − 1 ) . ( 1 ) C C C O O O C C C 0 0 0 l l l C C C M M M 7 已知椭圆C C C x 2 a 2 + y 2 b 2 = 1 ( a > b > 0 ) \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1(a>b>0) a 2 x 2 + b 2 y 2 = 1 ( a > b > 0 ) F F F x + y − 1 = 0 x+y-1=0 x + y − 1 = 0 2 2 . ( 1 ) \dfrac{\sqrt[]{2}}{2}.(1) 2 2 . ( 1 ) 8 已知过点P ( 1 , 3 2 ) P(1,\dfrac{3}{2}) P ( 1 , 2 3 ) C C C ( x − 1 ) 2 + y 2 + ( x + 1 ) 2 + y 2 = 2 a . ( Ⅰ ) \sqrt{{{(x-1)}^{2}}+{{y}^{2}}}+\sqrt{{{(x+1)}^{2}}+{{y}^{2}}}=2a.(Ⅰ) ( x − 1 ) 2 + y 2 + ( x + 1 ) 2 + y 2 = 2 a . ( Ⅰ ) C C C 9 设m ∈ R m∈R m ∈ R a → = ( m x , y + 1 ) \overrightarrow{a}=(mx,y+1) a = ( m x , y + 1 ) b → = ( x , y − 1 ) \overrightarrow{b}=(x,y-1) b = ( x , y − 1 ) a → ⊥ b → \overrightarrow{a}⊥ \overrightarrow{b} a ⊥ b 10 已知椭圆x 2 a 2 + y 2 b 2 = 1 ( a > b > 0 ) \dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0) a 2 x 2 + b 2 y 2 = 1 ( a > b > 0 ) F 1 ( − c , 0 ) F_{1}(-c,0) F 1 ( − c , 0 ) F 2 ( c , 0 ) F_{2}(c,0) F 2 ( c , 0 ) P P P a sin ∠ P F 1 F 2 = c sin ∠ P F 2 F 1 \dfrac {a}{\sin \angle PF_{1}F_{2}}= \dfrac {c}{\sin \angle PF_{2}F_{1}} sin ∠ P F 1 F 2 a = sin ∠ P F 2 F 1 c 11 设椭圆 C C C x 2 a 2 + y 2 b 2 = 1 ( a > b > 0 ) \dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}=1(a>b>0) a 2 x 2 + b 2 y 2 = 1 ( a > b > 0 ) F 1 F_{1} F 1 F 2 F_{2} F 2 2 2 \dfrac{\sqrt{2}}{2} 2 2 12 设点F 1 F_{1} F 1 F 2 F_{2} F 2 C : x 2 a 2 − y 2 2 = 1 ( a > 0 ) C: \dfrac{{x}^{2}}{{a}^{2}}- \dfrac{{y}^{2}}{2}=1\left(a > 0\right) C : a 2 x 2 − 2 y 2 = 1 ( a > 0 ) F 1 F_{1} F 1 x x x 13 已知F 1 F_{1} F 1 F 2 F_{2} F 2 C : x 2 a 2 − y 2 b 2 = 1 ( a > 0 , b > 0 ) C:\dfrac{{{x}^{2}}}{{{a}^{2}}}-\dfrac{{{y}^{2}}}{{{b}^{2}}}=1(a>0,b>0) C : a 2 x 2 − b 2 y 2 = 1 ( a > 0 , b > 0 ) P P P 14 设P P P x 2 a 2 − y 2 9 = 1 ( a > 0 ) \dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{9}=1(a>0) a 2 x 2 − 9 y 2 = 1 ( a > 0 ) 3 x − 2 y = 0 3x-2y=0 3 x − 2 y = 0 F 1 F_{1} F 1 F 2 F_{2} F 2 ∣ P F 1 ∣ = 3 |PF_{1}|=3 ∣ P F 1 ∣ = 3 15 已知双曲线C C C x 2 a 2 − y 2 b 2 = 1 ( a > 0 , b > 0 ) \dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1(a>0,b>0) a 2 x 2 − b 2 y 2 = 1 ( a > 0 , b > 0 ) F 1 F_{1} F 1 F 2 F_{2} F 2 M M M C C C I I I 16 已知点A ( 2 , 1 ) A(\sqrt{2},1) A ( 2 , 1 ) 2 2 \dfrac{\sqrt{2}}{2} 2 2 C C C x 2 a 2 + y 2 b 2 = 1 ( a > b > 0 ) \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1(a\text{ > }b\text{ > }0) a 2 x 2 + b 2 y 2 = 1 ( a > b > 0 ) 17 已知O O O M ( 1 , 2 ) M(1,2) M ( 1 , 2 ) P P P C C C y 2 = 2 p x ( p > 0 ) y^{2}=2px(p>0) y 2 = 2 p x ( p > 0 ) F F F F F F x 2 3 − y 2 = 1 \dfrac{{{x}^{2}}}{3}-{{y}^{2}}=1 3 x 2 − y 2 = 1 18 已知圆C 1 C_{1} C 1 x 2 + y 2 = r 2 ( r > 0 ) x^{2}+y^{2}=r^{2}(r > 0) x 2 + y 2 = r 2 ( r > 0 ) C 2 C_{2} C 2 x 2 = 2 p y ( p > 0 ) x^{2}=2py(p > 0) x 2 = 2 p y ( p > 0 ) ( 2 , − 2 ) ( \sqrt {2},-2) ( 2 , − 2 ) C 1 C_{1} C 1 C 2 C_{2} C 2 19 “m > − 1 m > -1 m > − 1 x 2 2 + m − y 2 1 + m = 1 \dfrac {x^{2}}{2+m}- \dfrac {y^{2}}{1+m}=1 2 + m x 2 − 1 + m y 2 = 1 20 把下列参数方程化为普通方程，并说明它们各表示什么曲线： ( 1 ) { x = 5 cos φ , y = 4 sin φ ( φ (1)\begin{cases} x=5\cos φ, \\ y=4\sin φ \end{cases}(φ ( 1 ) { x = 5 cos φ , y = 4 sin φ ( φ ) ) ) ( 2 ) { x = 1 − 3 t , y = 4 t ( t (2)\begin{cases} x=1-3t, \\ y=4t \end{cases}(t ( 2 ) { x = 1 − 3 t , y = 4 t ( t 21 若点P P P y = 3 y=3 y = 3 F ( 0 , − 2 ) F(0,-2) F ( 0 , − 2 ) 1 1 1 P P P ( ) () ( ) y 2 = 8 x y^{2}=8x y 2 = 8 x y 2 = − 8 x y^{2}=-8x y 2 = − 8 x x 2 = 8 y x^{2}=8y x 2 = 8 y x 2 = − 8 y x^{2}=-8y x 2 = − 8 y 22 已知关于x x x x 2 − 6 x + k + 7 = 0 x^{2}-6x+k+7=0 x 2 − 6 x + k + 7 = 0 k k k k k k 23 已知双曲线x 2 a 2 − y 2 b 2 = 1 ( a > 0 , b > 0 ) \dfrac{{{x}^{2}}}{{{a}^{2}}}-\dfrac{{{y}^{2}}}{{{b}^{2}}}=1(a > 0,b > 0) a 2 x 2 − b 2 y 2 = 1 ( a > 0 , b > 0 ) x 2 12 + y 2 4 = 1 \dfrac{{{x}^{2}}}{12}+\dfrac{{{y}^{2}}}{4}=1 12 x 2 + 4 y 2 = 1 24 过抛物线x 2 = 6 y x^{2}=6y x 2 = 6 y F F F A A A B B B M M M A B AB A B ( ) (\quad) ( ) A B AB A B y = − 3 2 y=-\dfrac{3}{2} y = − 2 3 25 已知双曲线χ 2 a 2 − γ 2 b 2 = 1 ( a > 0 , b > 0 ) \dfrac{{{\chi }^{2}}}{{{a}^{2}}}-\dfrac{{{\gamma }^{2}}}{{{b}^{2}}}=1(a > 0,b > 0) a 2 χ 2 − b 2 γ 2 = 1 ( a > 0 , b > 0 ) γ = 3 χ \gamma =\sqrt{3}\chi γ = 3 χ 26 双曲线x 2 4 − y 2 k = 1 \dfrac {x^{2}}{4}- \dfrac {y^{2}}{k}=1 4 x 2 − k y 2 = 1 e ∈ ( 1 , 2 ) e∈(1,2) e ∈ ( 1 , 2 ) k k k 27 “双曲线的方程为x 2 9 − y 2 16 = 1 \dfrac {x^{2}}{9}- \dfrac {y^{2}}{16}=1 9 x 2 − 16 y 2 = 1 y = ± 4 3 x y=± \dfrac {4}{3}x y = ± 3 4 x ( ) () ( ) 28 已知双曲线x 2 − y 2 3 = 1 x^{2}- \dfrac {y^{2}}{3}=1 x 2 − 3 y 2 = 1 M M M N N N y = x + m y=x+m y = x + m M N MN MN y 2 = 18 x y^{2}=18x y 2 = 18 x m m m 29 在一张纸上有一圆C C C ( x + 2 ) 2 + y 2 = r ( r > 0 ) (x+2)^{2}+y^{2}=r(r>0) ( x + 2 ) 2 + y 2 = r ( r > 0 ) M ( m , 0 ) ( m ≠ − 2 ) M(m,0)(m≠-2) M ( m , 0 ) ( m = − 2 ) C C C M ′ M' M ′ M M M P Q PQ PQ 30 过双曲线x 2 a 2 − y 2 b 2 = 1 ( a > 0 , b > 0 ) \dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1(a>0,b>0) a 2 x 2 − b 2 y 2 = 1 ( a > 0 , b > 0 ) F F F M M M F M FM FM A A A e e e 31 F F F C C C y 2 = 4 x y^{2}=4x y 2 = 4 x M ( m , 4 ) M(m,4) M ( m , 4 ) C C C M F MF MF C C C N N N ∣ F N ∣ = ( ) |FN|=(\quad) ∣ FN ∣ = ( ) 5 4 \dfrac{5}{4} 4 5 10 3 \dfrac{10}{3} 3 10 32 设双曲线的一个焦点为F F F B B B F B FB FB 2 \sqrt{2} 2 3 \sqrt{3} 3 1 + 3 2 \dfrac{1+\sqrt{3}}{2} 2 1 + 3 33 某双曲线的一条渐近方程为y = 3 2 x y=\dfrac{3}{2}x y = 2 3 x ( 0 , 26 ) (0,\sqrt{26}) ( 0 , 26 ) ( ) (\quad) ( ) x 2 6 − y 2 4 = 1 \dfrac{{{x}^{2}}}{6}-\dfrac{{{y}^{2}}}{4}=1 6 x 2 − 4 y 2 = 1 34 椭圆3 x 2 + 4 y 2 = 12 3x^{2}+4y^{2}=12 3 x 2 + 4 y 2 = 12 ( ) (\quad) ( ) ( ± 1 , 0 ) (±1,0) ( ± 1 , 0 ) ( 0 , ± 1 ) (0,±1) ( 0 , ± 1 ) ( ± 7 , 0 ) (±\sqrt{7},0) ( ± 7 , 0 ) ( 0 , ± 7 ) (0,±\sqrt{7}) ( 0 , ± 7 ) 35 写出适合条件的双曲线的标准方程： ( 1 ) a = 3 (1)a=3 ( 1 ) a = 3 b = 4 b=4 b = 4 x x x ( 0 , 5 ) (0,5) ( 0 , 5 ) ( 0 , − 5 ) (0,-5) ( 0 , − 5 ) ( 2 , 3 5 2 ) . (2, \dfrac {3 \sqrt {5}}{2}). ( 2 , 2 3 5 ) . 36 若椭圆x 2 a 2 + y 2 b 2 = 1 ( a > b > 0 ) \dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}=1\left( a > b > 0 \right) a 2 x 2 + b 2 y 2 = 1 ( a > b > 0 ) F F F P P P P P P 37 P P P A : ( x + 2 ) 2 + y 2 = 36 A:(x+2)^{2}+y^{2}=36 A : ( x + 2 ) 2 + y 2 = 36 B B B ( 2 , 0 ) (2,0) ( 2 , 0 ) P B PB PB A P AP A P Q Q Q Q Q Q C . ( 1 ) C.(1) C . ( 1 ) C C C ; ( 2 ) ;(2) ; ( 2 ) 38 已知椭圆C : x 2 a 2 + y 2 b 2 = 1 ( a > b > 0 ) \text{C}:\dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}=1(a>b>0) C : a 2 x 2 + b 2 y 2 = 1 ( a > b > 0 ) F ( 1 , 0 ) F(1,0) F ( 1 , 0 ) O O O C C C 39 已知方程x 2 2 − k + y 2 2 k − 1 = 1 \dfrac{x^{2}}{2-k}+\dfrac{y^{2}}{2k-1}=1 2 − k x 2 + 2 k − 1 y 2 = 1 y y y k k k ( ) (\quad) ( ) ( 1 2 , 2 ) (\dfrac{1}{2},2) ( 2 1 , 2 ) ( 1 , + ∞ ) (1,+\infty) ( 1 , + ∞ ) 40 若m > n > 0 m > n > 0 m > n > 0 m x 2 + n y 2 = 1 mx^{2}+ny^{2}=1 m x 2 + n y 2 = 1 ( ) () ( ) X X X y y y 41 设复平面上点对应的复数z = x + y i ( x ∈ R , y ∈ R ) ( i z=x+yi(x∈R,y∈R)(i z = x + y i ( x ∈ R , y ∈ R ) ( i ) ) ) ∣ z + 2 ∣ + ∣ z − 2 ∣ = 6 |z+2|+|z-2|=6 ∣ z + 2∣ + ∣ z − 2∣ = 6 Z Z Z C 1 . C_{1}. C 1 . C 2 C_{2} C 2 x 2 − y 2 n = 1 x^{2}- \dfrac {y^{2}}{n}=1 x 2 − n y 2 = 1 42 方程x 2 22 − m + y 2 6 + m = 1 \dfrac {x^{2}}{22-m} + \dfrac {y^{2}}{6+m} =1 22 − m x 2 + 6 + m y 2 = 1 y y y m m m 43 从抛物线x 2 = 4 y x^{2}=4y x 2 = 4 y P P P M M M ∣ P M ∣ = 5 |PM|=5 ∣ PM ∣ = 5 F F F △ M P F \triangle MPF △ MPF 44 若双曲线的中心为原点，F ( 0 , − 2 ) F(0,-2) F ( 0 , − 2 ) F F F l l l M M M N N N M N MN MN P ( 3 , 1 ) P(3,1) P ( 3 , 1 ) ( ) (\quad) ( ) x 2 3 − y 2 = 1 \dfrac{x^{2}}{3}-y^{2}=1 3 x 2 − y 2 = 1 45 设抛物线y 2 = 8 x y^{2}=8x y 2 = 8 x F F F P P P 4 4 4 ∣ P F ∣ |PF| ∣ PF ∣ ( ) () ( ) 8 8 8 6 6 6 4 4 4 2 2 2 46 对于椭圆x 2 4 + y 2 3 = 1 \dfrac{{{x}^{2}}}{4}+\dfrac{{{y}^{2}}}{3}=1 4 x 2 + 3 y 2 = 1 ( ) (\quad) ( ) 2 2 2 3 3 3 1 1 1 1 2 \dfrac{1}{2} 2 1 47 已知命题p p p q q q ( ) (\quad) ( ) p ∧ q p∧q p ∧ q p ∨ q p\lor q p ∨ q 48 已知抛物线C C C y 2 = 2 p x ( p > 0 ) y^{2}=2px(p > 0) y 2 = 2 p x ( p > 0 ) M M M ( x − 2 ) 2 + y 2 = 4 (x-2)^{2}+y^{2}=4 ( x − 2 ) 2 + y 2 = 4 M M M 3 3 3 P ( x 0 , y 0 ) ( x 0 ⩾ 5 ) P(x_{0},y_{0})(x_{0}\geqslant 5) P ( x 0 , y 0 ) ( x 0 ⩾ 5 ) 49 已知椭圆C ： x 2 3 + y 2 2 = 1 C： \dfrac {x^{2}}{3}+ \dfrac {y^{2}}{2}=1 C ： 3 x 2 + 2 y 2 = 1 P P P A ( − 3 , 0 ) A(- \sqrt {3},0) A ( − 3 , 0 ) B ( 3 , 0 ) B( \sqrt {3},0) B ( 3 , 0 ) ( ( ( ) ) ) 50 已知过抛物线y 2 = 2 p x ( p > 0 ) y^{2}=2px(p > 0) y 2 = 2 p x ( p > 0 ) 1 1 1 A A A B B B ∣ A B ∣ = 8 |AB|=8 ∣ A B ∣ = 8 51 已知椭圆C C C x 2 a 2 + y 2 b 2 = 1 ( a > b > 0 ) \dfrac{{x}^{2}}{{a}^{2}}+\dfrac{{y}^{2}}{{b}^{2}}=1(a>b>0) a 2 x 2 + b 2 y 2 = 1 ( a > b > 0 ) A A A B B B ∣ A B ∣ = 4 |AB|=4 ∣ A B ∣ = 4 2 2 . ( Ⅰ ) \dfrac{\sqrt{2}}{2}.(Ⅰ) 2 2 . ( Ⅰ ) 52 已知点P ( x 0 , y 0 ) P(x_{0},y_{0}) P ( x 0 , y 0 ) C 1 C_{1} C 1 C 2 C_{2} C 2 y 2 = 4 x y^{2}=4x y 2 = 4 x A A A B B B P A PA P A P B PB PB C 2 C_{2} C 2 A A A 53 过点P ( 2 , − 1 ) P(2,-1) P ( 2 , − 1 ) C C C x 2 = 2 y x^{2}=2y x 2 = 2 y A A A B B B A B AB A B 54 已知双曲线x 2 a 2 − y 2 b 2 = 1 ( a > 0 , b > 0 ) \dfrac{{x}^{2}}{{a}^{2}}-\dfrac{{y}^{2}}{{b}^{2}}=1(a{\rm\;>\;}0,b{\rm\;>\;}0) a 2 x 2 − b 2 y 2 = 1 ( a > 0 , b > 0 ) A A A F F F B 1 ( 0 , b ) B_{1}(0\text{,}b) B 1 ( 0 , b ) 55 已知椭圆x 2 3 + y 2 4 = 1 \dfrac{x^{2}}{3}+\dfrac{y^{2}}{4}=1 3 x 2 + 4 y 2 = 1 ( 1 , 1 ) (1,1) ( 1 , 1 ) ( ) (\quad) ( ) 4 x + 3 y − 7 = 0 4x+3y-7=0 4 x + 3 y − 7 = 0 4 x − 3 y − 7 = 0 4x-3y-7=0 4 x − 3 y − 7 = 0 56 已知椭圆C 1 C_{1} C 1 x 2 a 2 + y 2 b 2 = 1 ( a > b > 0 ) \dfrac{x^{2}}{a^{2}}{+}\dfrac{y^{2}}{b^{2}}{=}1(a{>}b{>}0) a 2 x 2 + b 2 y 2 = 1 ( a > b > 0 ) C 2 C_{2} C 2 x 2 − y 2 3 = 1 x^{2}{-}\dfrac{y^{2}}{3}{=}1 x 2 − 3 y 2 = 1 57 过抛物线x 2 = 4 y x^{2}=4y x 2 = 4 y P P P 1 2 \dfrac {1}{2} 2 1 P P P F F F ( ) () ( ) 1 1 1 2 2 2 58 设P P P x 2 a 2 + y 2 b 2 = 1 ( a > b > 0 ) \dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0) a 2 x 2 + b 2 y 2 = 1 ( a > b > 0 ) F 1 F_{1} F 1 F 2 F_{2} F 2 ∣ P F 1 ∣ + ∣ P F 2 ∣ = 4 |PF_{1}|+|PF_{2}|=4 ∣ P F 1 ∣ + ∣ P F 2 ∣ = 4 59 已知实数x x x y y y x 2 3 + y 2 = 1 \dfrac{x^{2}}{3}+y^{2}=1 3 x 2 + y 2 = 1 u = ∣ 3 x + 3 y − 7 ∣ u=|3x+3y-7| u = ∣3 x + 3 y − 7∣ 60 设双曲线C C C x 2 a 2 − y 2 b 2 = 1 ( a > 0 , b > 0 ) \dfrac{{{x}^{2}}}{{{a}^{2}}}-\dfrac{{{y}^{2}}}{{{b}^{2}}}=1\left(a>0,b>0\right) a 2 x 2 − b 2 y 2 = 1 ( a > 0 , b > 0 ) F 1 {{F}_{1}} F 1 F 2 {{F}_{2}} F 2