plzz answer this question..by steps
Answers
Answer:
(3) -35 < m < 15
Step-by-step explanation:
Let (x,y) be a point that is on both the circle and the line (i.e. a point of intersection).
Then the coordinates of this point satisfy both of the equations simultaneously:
x² + y² - 4x - 8y - 5 = 0 ... (1)
3x - 4y = m ... (2)
So to "find" (x,y), we need to solve this system of simultaneous equations.
Let's eliminate y to just have an equation for x.
From (2), we get
4y = 3x - m ... (3)
and multiplying (1) by 16 (just to avoid working with fractions), we get
16x² + (4y)² - 64x -32(4y) - 80 = 0 ... (4).
Substituting (3) into (4) gives
16x² + (3x-m)² - 64x - 32(3x-m) - 80 = 0
=> 16x² + 9x² - 6mx + m² - 64x - 96x + 32m - 80 = 0
=> 25x² - (6m + 160)x + (m² + 32m - 80) = 0
We don't actually need to find the point (x,y), we just need to say when there is more than one solution, since this corresponds to there being two distinct points of intersection.
What we have is a quadratic equation. This has more than one solution when the discriminant is positive. So...
The circle and line meet in two distinct points
<=> the discriminant of the quadratic equation above is positive
<=> (6m+160)² - 4(25)(m²+32m-80) > 0
<=> (3m+80)² - 25(m²+32m-80) > 0
<=> 9m² + 480m + 6400 - 25m² - 800m + 2000 > 0
<=> -16m² - 320m + 8400 > 0
<=> 16m² + 320m - 8400 < 0
<=> m² + 20m - 525 < 0
<=> ( m + 35 ) ( m - 15 ) < 0
<=> -35 < m < 15 [ as these values of m make one factor positive and the other negative ].
Answer:
Step-by-step explanation:
