A curve has equation y = 2 x y + 5 and a line has equation 2 x + 5 y = 1 . The curve and the line intersect at the points A and B . Find the coordinates of the midpoint of the line A B .

Given curve is y = 2 x y + 5 ...(i) Given line is 2 x + 5 y = 1 ...(ii) From (ii), x = 1 - 5 y 2 ⇒ x = 1 - 5 y 2 ⇒ 5 y 2 - 5 = 0 ⇒ 5 ( y - 1 ) ( y + 1 ) = 0 ⇒ y = - 1 y = - 1 ⇒ x = 3 y = 1 ⇒ x = - 2 The graph intersect at ( - 2 , 1 ) and ( 3 , - 1 ) The midpoint of the line joining these two points is - 2 + 3 2 , 1 - 1 2 Therefore, the coordinates of the midpoint of the line A B = 1 2 , 0

Find the real roots of the equation 36 x 4 + 4 = 25 x 2 .

Given that, 36 x 4 + 4 = 25 x 2 Multiplying each term with x 4 , ⇒ 36 + 4 x 4 = 25 x 2 ⇒ 4 x 4 - 25 x 2 + 36 = 0 ...(i) Let, y = x 2 , substitute in (i) ⇒ 4 y 2 - 25 y + 36 = 0 ⇒ ( 4 y - 9 ) ( y - 4 ) = 0 ⇒ y = 9 4 or y = 4 ⇒ x 2 = 9 4 or x 2 = 4 ∴ The real roots are x = ± 3 2 or x = ± 2

E M B I B E

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 1 - Quadratics>EXERCISE 1I>Q 13

MEDIUM

AS and A Level

IMPORTANT

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The line $y = m x + c$ is a tangent to the curve $a x^{2} + b y^{2} = c$ , where $a, b, c$ and $m$ are constants. Prove that $m^{2} = \frac{a b c - a}{b}$ .

Important Questions on Quadratics

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 1 - Quadratics>END-OF-CHAPTER REVIEW EXERCISE 1>Q 1

A curve has equation $y = 2 x y + 5$ and a line has equation $\Rightarrow - 4 a b c^{2} + 4 a c + 4 b m^{2} c = 0$ . The curve and the line intersect at the points $A$ and $B$ . Find the coordinates of the midpoint of the line $A B$ .